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Chapter 7: Problem 5

Show that the mapping $$ \left(w_{1}, \omega_{2}\right) \rightarrow w_{1} \wedge w_{2} $$ is bilinear and skew symmetric: $$ \begin{gathered} \omega_{1} \wedge \omega_{2}=-\omega_{2} \wedge \omega_{1} \\ \left(\lambda^{\prime}\left(\omega_{1}^{\prime}+\lambda^{\prime \prime} \omega_{1}^{\prime \prime}\right) \wedge \omega_{2}=\lambda^{\prime} \omega_{1}^{\prime} \wedge \omega_{2}+\lambda^{N} \omega_{1}^{\prime \prime} \wedge \omega_{2}\right. \end{gathered} $$

Short Answer

Expert verified
In conclusion, the given mapping $$\left(w_{1}, \omega_{2}\right) \rightarrow w_{1} \wedge w_{2}$$ is bilinear and skew symmetric because it satisfies the properties of skew symmetry, $$\omega_{1} \wedge \omega_{2}=-\omega_{2} \wedge \omega_{1}$$, and bilinearity, $$\left(\lambda^{\prime}\left(\omega_{1}^{\prime}+\lambda^{\prime \prime}\omega_{1}^{\prime \prime}\right) \wedge \omega_{2}=\lambda^{\prime}\omega_{1}^{\prime} \wedge \omega_{2}+\lambda^{N} \omega_{1}^{\prime \prime}\wedge \omega_{2}\right)$$, as demonstrated in the provided step-by-step solution.

Step by step solution

01

Show skew symmetry

To show skew symmetry, we need to prove that swapping the position of $$\omega_1$$ and $$\omega_2$$ introduces a negative sign: $$ \omega_{1} \wedge \omega_{2}=-\omega_{2} \wedge \omega_{1} $$ Since the wedge product is defined as an antisymmetric operation: $$ \omega_1 \wedge \omega_2 = -( \omega_2 \wedge \omega_1), $$ we have shown that the mapping is skew symmetric.
02

Show bilinearity

To show bilinearity, we need to prove that the following equality holds: $$ \left(\lambda^{\prime}\left(\omega_{1}^{\prime}+\lambda^{\prime \prime}\omega_{1}^{\prime \prime}\right) \wedge \omega_{2}=\lambda^{\prime}\omega_{1}^{\prime} \wedge \omega_{2}+\lambda^{N} \omega_{1}^{\prime \prime}\wedge \omega_{2} $$ Let's prove this by using the definition of the wedge product: \begin{align*} \left(\lambda^{\prime}\left(\omega_{1}^{\prime}+\lambda^{\prime \prime}\omega_{1}^{\prime \prime}\right) \wedge \omega_{2}\right) &= \left( \lambda^{\prime}\omega_{1}^{\prime} + \lambda^{\prime \prime}\omega_{1}^{\prime \prime} \right) \wedge \omega_{2} \\ &= \lambda^{\prime} \left( \omega_{1}^{\prime} \wedge \omega_{2} \right) + \lambda^{\prime \prime} \left( \omega_{1}^{\prime \prime} \wedge \omega_{2} \right) \\ &= \lambda^{\prime}\omega_{1}^{\prime} \wedge \omega_{2}+\lambda^{N} \omega_{1}^{\prime \prime}\wedge \omega_{2}. \end{align*} Hence, the bilinearity property is satisfied. By proving both skew symmetry and bilinearity, we have shown that the given mapping is indeed bilinear and skew symmetric.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bilinear Mapping
A bilinear mapping, in the context of vector spaces, is an operation that accepts two vectors and produces another vector or scalar, satisfying two key properties. These properties are additivity and scalar compatibility. Additivity means that the operation is linear in each of its arguments separately. When it comes to scalar compatibility, if you have a scalar multiple of a vector, this scalar can be pulled out and multiplied with the result after the operation is performed.

For instance, in the exercise, the wedge product is shown to be a bilinear mapping because it adheres to these properties. The wedge product \( \omega_1 \wedge \omega_2 \) can distribute over addition and can factor out scalar multiples, as evidenced in the steps of the solution. Through distributing and factoring out scalars, the operation shows bilinearity - an essential concept in vector algebra and many areas of mathematics and physics.
Skew Symmetry
Skew symmetry, sometimes called antisymmetry, is a characteristic of certain operations in which the order of the inputs affects the sign of the output. Mathematically, an operation \( * \) is skew symmetric if swapping the inputs negates the result: \( a * b = -(b * a) \).

In our exercise, the wedge product exhibits this property, meaning that the operation changes sign whenever the order of the vectors is reversed. This concept is foundational in forms and algebraic structures such as Lie algebras, and it is intrinsic to areas such as differential forms in calculus.
Antisymmetric Operation
An antisymmetric operation is one in which the result changes sign whenever two inputs are swapped, and this is a stricter form of antisymmetry where the operation results in zero when both inputs are the same. In the exercise, the wedge product between two identical vectors would yield zero, which is an essential aspect of antisymmetry. This behavior aligns with the intuitive notion of determinants in linear algebra, where flipping two identical rows results in the determinant being zero.

Antisymmetric operations are widely used to model various phenomena in physics, including electromagnetic fields and quantum mechanics, where the orientation of inputs relative to each other is crucial.
Classical Mechanics
Classical mechanics is the branch of physics that deals with the motion of bodies based on Isaac Newton's laws of motion. It is fundamental to understanding the everyday physical world. Concepts such as force, energy, momentum, and torque are rooted in classical mechanics. The wedge product enters classical mechanics in a more abstract form, often used in the study of angular momentum, where the orientation and amount of rotational momentum depend on the order and magnitude of the position and force vectors.

Exploring the wedge product in this field offers a more profound insight into phenomena like the gyroscopic effect and conservation laws, which are keystones in the study of mechanical systems. Understanding how bilinear and antisymmetric properties play roles in physics helps to deepen comprehension in topics like rigid body motion and orbital mechanics.

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Most popular questions from this chapter

Show that the maps \(\mathbf{A} \rightarrow \omega_{A}^{1}\) and \(\mathrm{A} \rightarrow \omega_{A}^{2}\) establish isomorphisms of the lineat spacx \(\mathbb{R}^{3}\) of vectors \(A\) with the linear spaces of 1 -forms on \(\mathbb{R}^{3}\) and 2 -forms on \(\mathbb{R}^{3}\). If we choose an orthonormal oriented coordinate system \(\left(x_{1}, x_{2}, x_{3}\right)\) on \(\mathbb{R}^{3}\). then $$ \omega_{A}^{1}=A_{1} x_{1}+A_{2} x_{2}+A_{3} x_{3} $$

Let \(x_{1}, \ldots, x_{n}\) be functions on a manifold \(M\) forming a local coordinate system in some region. Show that every 1-form on this region can be uniquely written in the form \(\omega=a_{1}(x) d x_{1}+\cdots+a_{e}(x) d x_{n}\).

Show that the \(k\)-forms on \(M\) form a vector space (infinite-dimensional if \(k\) does not exceed the dimension of \(M\) ).

Show that integration of a fixed form \(\omega^{k}\) on chains \(c_{k}\) defines a homomorphism from the group of chains to the line.

Show that for the circle \(S^{1}\) we have \(H^{1}\left(S^{1}, \mathbb{R}\right)=\mathbb{R}\).
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